series solution of unsteady free convection flow with mass transfer along an accelerated vertical...

Cent. Eur. J. Phys. • 8(6) • 2010 • 931-939DOI: 10.2478/s11534-010-0007-y

Central European Journal of Physics

Series solution of unsteady free convection flow withmass transfer along an accelerated vertical porousplate with suction

Research Article

Haider Zaman1∗, Muhammad Ayub2

1 Department of Mathematics, Islamia College Chartered University 25120, Peshawar 25000, Pakistan

2 Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

Received 27 September 2009; accepted 23 December 2009

Abstract: The problem of unsteady free convection flow is considered for the series solution (analytic solution). Theflow is induced by an infinite vertical porous plate which is accelerated in its own plane. The series solutionexpressions for velocity field, temperature field and concentration distribution are presented. The influenceof important parameters is seen on the velocity, temperature, concentration, skin friction coefficient andtemperature gradient with the help of graphs and tables. Convergence is also properly checked for differentvalues of the important parametes for velocity field, temperature and concentration with the help of ~-curves.

PACS (2008): 47.10.A-, 47.10.ad, 47.15.Cb, 47.50.Cd, 02.70.Wz

Keywords: unsteady • mass transfer • porosity • series solution© Versita Sp. z o.o.

1. Introduction

Boundary layer flows of viscous fluids induced by an ac-celerated infinite vertical porous flat plate have receivedconsiderable attention due to their many technologicaland industrial applications, particularly in the field ofcosmical and geophysical sciences. Since the pioneer-ing work of Sakiadis [1] such flows have been studiedextensively through various aspects. The literature onthe topic is quite abundant. Some recent contributionshave been made by Hayat et al. [2–10], Asghar et al. [11–13], Raptis et al. [14–17], Bataller [18], Cortell [19, 20],∗E-mail: [email protected]

Ariel et al. [21, 22] and Khan et al. [23–25]. Hayat etal. [2–10] have studied flows of second grade, third grade,fourth grade, Oldroyd 6-constant, micropolar and JohnsonSegalman fluids for various situations with and withoutheat transfer analysis, and with and without magnetohy-drodynamics effects. Asghar et al. [11–13] have discussedflows of non-Newtonian fluids with Hall effect and heattransfer. Raptis et al. [14–17] have investegated free con-vection flows under different conditions. Bataller [18] dis-cussed effects of heat on flows of a viscoelastic fluids overa stretching sheet. Cortell [19, 20] also studied flow overa stretching sheet. Ariel et al. [21, 22] discussed flow overstretching sheet with partial slip. Khan et al. [23–25] havediscussed flows of Sisko and Burgers’ fluids with Hall ef-fect.

931

Series solution of unsteady free convection flow with mass transfer along an accelerated vertical porous plate with suction

Das et al. [26] have considered numerical solution of masstransfer effects on unsteady flow past an accelerated ver-tical porous plate with suction. The purpose of this com-munication is to revisit the problem discussed in case ofDas et al. [26] for the series solution with HAM. HAM is areliable method which has been already used by severalresearchers [27–44] in finding the series solutions of manyproblems.2. Series solution for the velocityfield, temperature field and concen-tration distributionThe problem in [26] is

f ′′ + 2 (η+ a) f ′ − αf = −Gr θ − GcC, (1)θ′′ + 2 (η+ a)Pr θ′ = 0, (2)C ′′ + 2 (η+ a)Sc C ′ = 0, (3)f(0) = 1, f(∞) = 0, (4)θ (0) = 1, θ (∞) = 0, (5)C (0) = 1, C (∞) = 0. (6)

where prime indicates the differentiation with respect toη only, a > 0 is the suction parameter, α is the porosityparameter, Gr is the Grashof number for heat transfer, Gcis the Grashof number for mass transfer, Pr is the Prandtlnumber, Sc is the Schmidt number, f is the velocity field,θ is the temperature field and C is the concentration dis-tribution.2.1. Zeroth-order deformation problemsFor series solutions the initial approximations of f (η), θ (η)and C (η) are taken in such a wary that they must sat-isfy the boundary conditions. The boundary conditions inEqs. (4), (5) and (6) leads us to assume following initialapproximations

f0 (η) = e−η, (7)θ0 (η) = e−η, (8)C0 (η) = e−η, (9)

and the auxiliary linear operators areL1(f) = f ′′ − f, (10)L2(θ) = θ′′ − θ, (11)

L3(C ) = C ′′ − C, (12)L1 [C1eη + C2e−η] = 0, (13)L2 [C3eη + C4e−η] = 0, (14)L3 [C5eη + C6e−η] = 0, (15)

where Ci (i = 1− 6) are arbitrary constants.From Eqs. (1), (2) and (3) we define the non-linear oper-atorsN1 [f(η, p)] = ∂2 f(η, p)

∂η2 + 2 (η+ a) ∂f(η, p)∂η

− αf(η, p) + Gr θ(η, p) + GcC (η, p), (16)N2 [θ(η, p)] = ∂2θ(η, p)

∂η2 + 2 (η+ a)Pr ∂θ(η, p)∂η , (17)

N3 [θ(η, p)] = ∂2C (η, p)∂η2 + 2 (η+ a)Sc ∂C (η, p)

∂η , (18)and then construct the following zeroth order problems

(1− p) L1 [f(η, p)− f0 (η)] = p ~1 N1 [f(η, p)] , (19)f (0, p) = 1, f (∞, p) = 0, (20)

(1− p) L2 [θ(η, p)− θ0 (η)] = p ~2 N2 [θ(η, p)] , (21)θ (0, p) = 1, θ (∞, p) = 0, (22)

(1− p) L3 [C (η, p)− C0 (η)] = p ~3 N3 [C (η, p)] , (23)C (0, p) = 1, C (∞, p) = 0, (24)

where ~i (i = 1, 2, 3) are the non-zero auxiliary parame-ters and p ∈ [0, 1] is an embedding parameter. For p = 0and p = 1, one may writef (η, 0) = f0 (η) , f (η, 1) = f(η), (25)θ (η, 0) = θ0 (η) , θ (η, 1) = θ(η), (26)C (η, 0) = C0 (η) , C (η, 1) = C (η). (27)

As the embedding parameter p increases from 0 to 1,f (η, p), θ(η, p) and C (η, p) varies (or deforms) from f0 (η),θ0 (η) and C0 (η) to f (η), θ(η) and C (η) respectively.Through Taylor’s theorem and Eqs. (25), (26) and (27),one can write

f (η, p) = f0 (η) + ∞∑m=1 fm (η) pm, (28)

932

Haider Zaman, Muhammad Ayub

θ (η, p) = θ0 (η) + ∞∑m=1 θm (η) pm, (29)

C (η, p) = C0 (η) + ∞∑m=1 Cm (η) pm, (30)

fm (η) = 1m! ∂m f (η, p)

∂pm

∣∣∣∣∣p=0 , (m ≥ 1) , (31)

θm (η) = 1m! ∂mθ (η, p)

∂pm

∣∣∣∣∣p=0 , (m ≥ 1) , (32)

Cm (η) = 1m! ∂mC (η, p)

∂pm

∣∣∣∣∣p=0 , (m ≥ 1) . (33)

Obviously, the convergence of the series (28), (29) and(30) depends upon ~1, ~2 and ~3. Assume that ~1, ~2 and~3 are chosen such that the series (28), (29) and (30) areconvergent at p = 1, then we obtain

f (η) = f0 (η) + ∞∑m=1 fm (η) , (34)

θ (η) = θ0 (η) + ∞∑m=1 θm (η) , (35)

C (η) = C0 (η) + ∞∑m=1 Cm (η) . (36)

2.2. mth-order deformation problemsThe mth− order deformation problems are

L1 [fm (η)− χmfm−1 (η)] = ~1 R1m(η), (37)fm (0) = 0, fm (∞) = 0, (38)

L2 [θm (η)− χmθm−1 (η)] = ~2 R2m(η), (39)θm (0) = 0, θm (∞) = 0, (40)

L3 [Cm (η)− χmCm−1 (η)] = ~3 R3m(η), (41)Cm (0) = 0, Cm (∞) = 0, (42)

R1m(η) = f ′′m−1 + 2 (η+ a) f ′m−1 − αfm−1+ Grθm−1 + GcCm−1, (43)R2m(η) = θ′′m−1 (η) + 2 (η+ a)Prθ′m−1, (44)R3m(η) = C ′′m−1 (η) + 2 (η+ a)ScC ′m−1, (45)

χm = { 0, m ≤ 1,1, m ≥ 2. (46)By means of symbolic computation software MATHEMAT-ICA, the solution of above problems are

fm(η) = 2m∑n=0 a1m,nηne−η, m ≥ 0, (47)

θm (η) = 2m∑n=0 a2m,nηne−η, m ≥ 0, (48)

Cm (η) = 2m∑n=0 a3m,nηne−η, m ≥ 0. (49)

Invoking Eq. (47) into Eq. (37), Eq. (48) into Eq. (39) andEq. (49) into Eq. (41), we get the following recurrenceformulae for the coefficients a1m,n, a2m,n and a3m,n offm(η), θm (η) and Cm (η) respectively as follows for m ≥ 1,0 ≤ n ≤ 2m.

a1m,1 = −α1m,02 + χm a1m−1,1 −2m−2∑n=0 α1m,nµn,0, (50)



a1m,n = χm a1m−1,n −2m−2∑k=n−1 α1m,kµk,n−1, n ≥ 2, (53)



α1m,0 = ~1(c1m−1,0 + 2ab1m−1,0 − αa1m−1,0+Gra2m−1,0 + Gca3m−1,0

), (56)

α2m,0 = ~2 (c2m−1,0 + 2aPrb2m−1,0) , (57)α3m,0 = ~3 (c3m−1,0 + 2aScb3m−1,0) , (58)

α1m,n = ~1(c1m−1,n + 2ab1m−1,n − αa1m−1,n + Gra2m−1,n+Gca3m−1,n + 2b1m−1,n−1

),

(59)933


α2m,n = ~2 (c2m−1,n + 2aPrb2m−1,n+2Prb2m−1,n−1) , (60)α3m,n = ~3 (c3m−1,n + 2aScb3m−1,n+2Scb3m−1,n−1) , (61)

µn,k = n!(k + 1)!2n−k+1 (62)b1m,n = (n+ 1)a1m,n+1 − a1m,n, (63)b2m,n = (n+ 1)a2m,n+1 − a2m,n, (64)b3m,n = (n+ 1)a3m,n+1 − a3m,n, (65)c1m,n = (n+ 1)b1m,n+1 − b1m,n, (66)c2m,n = (n+ 1)b2m,n+1 − b2m,n, (67)c3m,n = (n+ 1)b3m,n+1 − b3m,n. (68)

The corresponding Mth order approximation of Eqs. (1)-(4), (2)-(5) and (3)-(6) areM∑m=0 fm(η) = M∑

m=0( 2m∑

n=0 a1m,nηn) e−η, (69)M∑m=0 θm (η) = M∑

m=0( 2m∑

n=0 a2m,nηn) e−η, (70)M∑m=0 Cm (η) = M∑

m=0( 2m∑

n=0 a3m,nηn) e−η, (71)and the analytic solution aref(η) = ∞∑

m=0 fm(η) = limM→∞

[ M∑m=0( 2m∑

n=0 a1m,nηn) e−η] , (72)θ (η) = ∞∑

m=0 θm (η) = limM→∞

[ M∑m=0( 2m∑

n=0 a2m,nηn) e−η] ,(73)C (η) = ∞∑

m=0 Cm (η) = limM→∞

[ M∑m=0( 2m∑

n=0 a3m,nηn) e−η] .(74)The coefficients a1m,n, a2m,n and a3m,n can be computedby usinga10,0 = 1, a20,0 = 1, a30,0 = 1. (75)

given by the initial guess approximations in Eqs. (7), (8)and (9).

3. Convergence of the analytic solu-tionClearly Eqs. (72), (73) and (74) contain the auxiliary pa-rameters ~1, ~2 and ~3 respectively, which give the conver-gence region and rate of approximation for the homotopyanalysis method. For this purpose, the ~-curves are plot-ted for f , θ and C . From Fig. 1 for f (η) we observe that therange for the admissible value for ~1 is −0.55 < ~1 < 0.From Fig. 2 for θ (η) the range for the admissible values for~2 is −1.6 < ~2 ≤ −0.1. Fig. 3 for C (η) shows the rangefor the admissible values for ~3 is −1.7 < ~3 ≤ −0.1. Tosee the valid values of ~1 for different values of α Fig. 4is plotted. Fig. 4 depicts that ~1 = −1.5 is valid for0 < α < 8, ~1 = −1 is valid for 0 < α < 10, ~1 = −0.7is valid for 0 < α < 13. Fig. 5 is plotted to see the validvalues of ~1 for different values of the parameter a. Fig. 5shows that ~1 = −1.5 is valid for 0 < a < 10, ~1 = −1 isvalid for 0 < a < 20, ~1 = −0.7 is valid for 0 < a < 30.Fig. 6 is drawn to look the valid values for ~2 for differ-ent values of the suction parameter a. Fig. 6 shows that~2 = −1.5 is valid for 0 < a < 50, ~2 = −1 is valid for0 < a < 100, ~2 = −0.7 is valid for 0 < a < 150. Fig. 7 isplotted to see the valid values of ~3 for different values ofthe parameter a. Fig. 7 shows that ~3 = −1.5 is valid for0 < a < 70, ~3 = −1 is valid for 0 < a < 80, ~3 = −0.7 isvalid for 0 < a < 130. The computations made shows thatthe series of the velocity field f (η) in Eq. (72) convergesin the whole region of η when ~1 = −0.03, ~2 = −0.3 and~3 = −0.3. The series of heat flux (73) converges in thewhole region of η when ~2 = −0.3. The series of C (η), (74)converges in the whole region of η when ~3 = −0.3.4. Results and discussionNow we discuss the variation of the horizontal velocitycomponent f (η) with distance from the surface η for differ-ent values of the suction parameter a, porosity parameterα , Grashof number for heat transfer Gr , Grashof numberfor mass transfer Gc , Prandtl number Pr , and the Schmidtnumber Sc . The graphs are plotted for the 15th order ofapproximations.Fig. 8 shows the variation of horizontal velocity componentf (η) with distance from the surface η for several values ofthe suction parameter a. It is observed that for the fixedvalues of α , Pr , Sc , Gr and Gc , with the increase in suc-tion parameter a the horizontal component of the velocitydecreases at all points. It is also observed that greatersuction leads to a faster reduction in the velocity field.Fig. 9 is plotted for several values of porosity parameterα and fixed values of a, Pr , Sc , Gr and Gc . From the Fig. 9

934


Figure 1. ~1-curve for the 15th order of approximations for the ve-locity field f(η) at ~2 = −0.3, ~3 = −0.3, a = 0.02, α = 1,Pr = 0.071, Sc = 0.022, Gr = 1, Gc = 1.

Figure 2. ~2-curve for the 22th order of approximations for the tem-perature field θ(η) at a = 0.02, Pr = 0.071.

Figure 3. ~3-curve for the 22th order of approximations for the con-centration distribution C (η) at a = 0.01, Sc = 0.022.

it is observed that with the increase in porosity parametervelocity field decreases at all points. Fig. 10 depicts theeffects of Gr and Gc on the horizontal component of thevelocity f (η) for α = 1, Pr = 0.71 and Sc = 0.22. It is ob-served that with the increase in Grashof number for heattransfer Gr and mass transfer Gc the horizontal velocitycomponent f (η) increases at all points. The comparisonof curves in Fig. 10 also shows that mass transfer has adominant effect on the velocity field f (η). Fig. 11 eluci-

Figure 4. α − ~1-curves for 15th order of approximations for thevelocity field f(η) at ~2 = −0.3, ~3 = −0.3, a = 0.1,Pr = 0.71, Sc = 0.22, Gr = 1, Gc = 1.

Figure 5. a − ~1-curves for 15th order of approximations for thevelocity field f(η) at ~2 = −0.3, ~3 = −0.3, α = 0.1,Pr = 0.71, Sc = 0.22, Gr = 1, Gc = 1.

Figure 6. a−~2-curves for 15th order of approximations for the tem-perature field θ(η) at Pr = 0.071.

dates the variation of horizontal velocity component f (η)with distance from the surface η for several values of theSchmidt number Sc for fixed values a = 0.1, Pr = 0.71,α = 1, Gr = 1 and Gc = 1. Fig. 11 shows a decrease invelocity at all points with increase in Sc .Graphs are plotted for the temperature distribution θ (η)for 22th order of approximations. These graphs show vari-ation of θ (η) with η for suction parameter a and Prandtl

935


Figure 7. a−~3-curves for 15th order of approximations for the con-centration distribution C (η) at Sc = 0.22.

number Pr .Fig. 12 shows that by keeping Prandtl number Pr fixed,with increase in suction parameter a decreases the tem-perature of the flow field at all points. Fig. 13 depicts thevariation of temperature field θ (η) with η for several val-ues of Prandtl number Pr and for a fixed value of suctionparameter a. It is observed that with the increase in Prtemperature of the flow field decreases at all points.Graphs are also plotted for the concentration distribu-tion C (η) for 22th order of approximations. These graphsshow variation of C (η) with η for suction parameter a andSchmidt number Sc .Fig. 14 elucidates the variation of C (η) with η for a fixedvalue of suction parameter a = 0.1 and several values ofthe Schmidt number Sc . It is observed that with increasein Schmidt number concentration distribution decreases.The higher values of the Schmidt number leads to a fasterdecrease in concentration of the flow field. Fig. 15 ex-plains that for a fixed value of the Schmidt number Sc theconcentration distribution decreases with increase in thesuction parameter a. The greater suction leads to a fasterdecrease in concentration of the flow field.Tab. 1 shows the values of skin friction coefficient −f ′(0)for 25th order of approximations when Pr = 0.71, Gr = 1,Gc = 1, Sc = 0.22, ~1 = −0.03, ~2 = −0.3 and ~3 =−0.3. It is observed from Tab. 1 that increase in suctionparameter a leads to increase in the value of skin frictionat the wall −f ′(0). It is also observed from Tab. 1 thatincrease in porosity parameter α leads to increase in thevalue of skin friction at the wall −f ′(0). Tab. 2 indicatesthe values of heat transfer rate at the wall −θ′(0) for 25thorder of approximations and Pr = 0.71, ~2 = −0.3. Weobserved from Tab. 2 that an increase in suction parametera yields an increase in heat flux at the wall −θ′(0).

Figure 8. Velocity profiles against η for different values of awith ~1 =−0.03, ~2 = −0.3, ~3 = −0.3, α = 1, Pr = 0.71, Sc = 0.22,Gr = 1, Gc = 1.

Figure 9. Velocity profiles against η for different values of α with ~1 =−0.03, ~2 = −0.3, ~3 = −0.3, a = 0.1, Pr = 0.71, Sc =0.22, Gr = 1, Gc = 1.

Figure 10. Velocity profiles against η for different values of Gr andGc with ~1 = −0.03, ~2 = −0.3, ~3 = −0.3, α = 1,Pr = 0.71, Sc = 0.22.

5. Final remarks

In this study the series solution for the velocity field,temperature field and concentration distribution are con-structed. Analytic solutions are more important than thenumerical solutions because they are valid on the wholedomain of the definition, where as numerical solutions are936


Figure 11. Velocity profiles against η for different values of Sc with~1 = −0.03, ~2 = −0.3, ~3 = −0.3, a = 0.1, Pr = 0.71,α = 1, Gr = 1, Gc = 1.

Figure 12. Temperature profiles against η for different values of awith ~2 = −0.1, Pr = 0.71.

Figure 13. Temperature profiles against η for different values of Prwith ~2 = −0.1, a = 0.1.

valid only at chosen points of the domain. The conver-gence of the series solution is also properly checked anddiscussed for different values of the important parameters.The graphical comparison of our analysis with those ofDas et al. [26] shows that our results are in good agree-ment with those of Das et al. [26]. The same type ofbehavior as in case of Das et al. [26] for velocity field,temperature field and concentration distribution are ob-

Figure 14. Concentration profiles against η for different values of Scwith ~3 = −0.1, a = 0.1.

Figure 15. Concentration profiles against η for different values of awith ~3 = −0.1, Sc = 0.22.

Table 1. Values of skin friction coefficient −f ′(0) for different valuesof a and α with Pr = 0.71, Gr = 1, Gc = 1, Sc = 0.22,~1 = −0.03, ~2 = −0.3 and ~3 = −0.3.

a −f ′(0) α −f ′(0)0.1 0.5625301 0.1 0.56253010.5 0.8171164 0.5 0.67625611 1.1735210 1 0.80941452 1.9997210 2 1.05506995 5.0520550 5 1.636705010 10.432930 10 2.3628690

Table 2. Values of heat transfer coefficient −θ′(0) for different valuesof a when Pr = 0.71 and ~2 = −0.3.

a −θ′(0)0.1 0.98643170.5 1.1658111 1.4087312 1.9528775 3.94434610 7.690454

937


served for different values of the important parameters.We may claim that our analysis in two ways better thanthose of Das et al. [26], one is our full solution for all theequations is analytic and second we have a proper wayof checking and controlling the convergence of the seriessolution. But in case of Das et al. [26] Eqs. (1) and (4) aresolved numerically and convergence is not discussed.We have following observations about the effects of perti-nent parameters in the flow field on the velocity, temper-ature, skin friction, rate of heat transfer and on concen-tration:• With the increase in suction parameter velocity ofthe flow field decreases at all points and whenthe suction parameter becomes larger than this de-crease in the velocity is faster.• Porosity parameter decelerates the velocity of theflow field at all points.• Grashof number for heat transfer and mass transferaccelerate the velocity of the flow field at all points.• With the increase in Schmidt number velocity of theflow field decreases at all points.• The temperature of the flow field decreases at allpoints with the increase in suction.• Larger suction leads to faster cooling of the fluidand plate.• With the increase in Prandtl number temperatureof the flow field decreases in magnitude.• Larger Schmidt number leads to faster decrease inconcentration of the flow field.• When porosity parameter becomes larger, the con-centration distribution of the flow field decreasesfaster.• With an increase in suction skin friction at the wallincreases.• With an increase in suction rate of heat transfer atthe wall increases.

AcknowledgementsOne of the author Dr. Haider Zaman is grateful to theanonymous referee for the useful comments.

References

[1] B. C. Sakiadis, AIChE J. 7, 26 (1961)[2] T. Hayat, Z. Abbas, M. Sajid, Phys. Lett. A. 372, 2400(2008)[3] T. Hayat, R. Ellahi, F. M. Mahomed, Commun. Non-linear Sci. 14, 133 (2009)[4] T. Hayat, R. Ellahi, F. M. Mahomed, Chaos Soliton.Fract. 38, 1336 (2008)[5] T. Hayat, H. Mambili-Mamboundou, F. Mahomed,Nonlinear Anal.-Real 10, 368 (2009)[6] T. Hayat, M. A. Farooq, T. Javed, M. Sajid, NonlinearAnal.-Real 10, 745 (2009)[7] T. Hayat, T. Javed, Z. Abbas, Nonlinear Anal.-Real 10,1514 (2009)[8] T. Hayat, Z. Iqbal, M. Sajid, K. Vajravelu, Int. Com-mun. Heat Mass 35, 1297 (2008)[9] T. Hayat, Z. Abbas, T. Javed, J. Porous Media 12, 183(2009)[10] T. Hayat, M. Umar Qureshi, Q. Hussain, Appl. Math.Model. 33, 1862 (2009)[11] S. Asghar, M. Khan, T. Hayat, Int. J. Nonlin. Mech.40, 589 (2005)[12] S. Asghar, T. Hayat, P. D. Ariel, Commun. NonlinearSci. 14, 154 (2009)[13] S. Asghar, R. Mohyuddin, T. Hayat, Int. J. Heat MassTran. 48, 599 ( 2005)[14] A. Raptis, G. Perdikis, G. Tzivanidis, J. Phys. D. 14,99 (1981)[15] A. Raptis, G. Tzivanidis, N. Kafousios, Lett. HeatMass Trans. 8, 417 (1981)[16] A. Raptis, N. Kafousios, C. Massalas, ZAMM-Z.Angew. Math. Me. 62, 489 (1982)[17] A. Raptis, Int. J. Eng. Sci. 21, 345 (1983)[18] R. C. Bataller, Comput. Math. Appl. 53, 305 (2007)[19] R. Cortell, Int. J. Nonlin. Mech. 41, 78 (2006)[20] R. Cortell, Chem. Eng. Process. 46, 721 (2007)[21] P. D. Ariel, T. Hayat, S. Asghar, Acta Mech. 187, 29(2006)[22] P. D. Ariel, T. Hayat, S. Asghar, Int. J. Nonlin. Sci.Num. 7, 399 (2006)[23] M. Khan, Z. Abbas, T. Hayat, Transport Porous Med.71, 23 (2008)[24] M. Khan, S. Hyder Ali, T. Hayat, C. Fetecau, Int. J.Nonlin. Mech. 43, 302 (2008)[25] M. Khan, S. Asghar, T. Hayat, Nonlinear Anal.-Real10, 974 (2009)[26] S. S. Das, S. K. Sahoo, G. C. Dash, B. Malays. Math.Sci. So. 29, 33 (2006)[27] S. J. Liao, Commun. Nonlinear Sci. 14, 983 (2009)[28] S. J. Liao, Jian Su, Allen T. Chwang, Int. J. Heat Mass938


Tran. 49, 2437 (2006)[29] S. J. Liao, Commun. Nonlinear Sci. 11, 326 (2006)[30] S. J. Liao, Appl. Math. Comput. 147, 499 (2004)[31] S. J. Liao, Int. J. Nonlin. Mech. 38, 1173 (2003)[32] S. J. Liao, A. Campo, J. Fluid Mech. 453, 411 (2002)[33] S. J. Liao, Int. J. Nonlin. Mech. 7, 1 (2002)[34] S. J. Liao, Int. J. Nonlin. Mech. 34, 759 (1999)[35] S. J. Liao, PhD thesis, Shanghai Jiao Tong University(Shanghai, China, 1992)[36] S. Abbasbandy, T. Hayat, R. Ellahi, S. Asghar, Z.Naturforsch. Pt. A 64, 59 (2009)[37] S. Abbasbandy, F. S. Zakaria, Nonlinear Dynam. 51,83 (2008)[38] S. Abbasbandy, Chem. Eng. J. 136, 144 (2008)[39] S. Abbasbandy, Phys. Lett. A. 361, 478 (2007)[40] S. Abbasbandy, Int. Commun. Heat Mass 34, 380(2007)[41] S. Abbasbandy, Phys. Lett. A. 360, 109 (2006)[42] M. Ayub, A. Rashid, T. Hayat, Int. J. Eng. Sci. 41, 2091(2003)[43] M. Ayub, H. Zaman, M. Sajid, T. Hayat, Commun.Nonlinear Sci. 13 1822 (2008)[44] M. Ayub, H. Zaman, M. Ahmad, Cent. Eur. J. Phys. 8,135 (2010)

939

series solution of unsteady free convection flow with mass transfer along an accelerated vertical...

Documents